theorem :: FDIFF_9:25

for Z being open Subset of REAL st Z c= dom (exp_R (#) cosec) holds
( exp_R (#) cosec is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cosec) `| Z) . x = ((exp_R . x) / (sin . x)) - (((exp_R . x) * (cos . x)) / ((sin . x) ^2)) ) )